category of sets
- notation:
- objects: sets
- morphisms: maps
- nLab Link
- Related categories: ,
The category of sets plays a fundamental role in category theory. Due to the Yoneda embedding, many results about general categories can be reduced to the category of sets. It is also usually the first example of a category that one encounters.
Properties
Properties from the database
- is a Grothendieck topos
- is finitary algebraic
- is locally small
Deduced properties
- is locally essentially small
- is locally finitely presentable
- is locally presentable
- is cocomplete
- is complete
- has connected limits
- has filtered limits
- is finitely complete
- has wide pullbacks
- has equalizers
- has products
- has finite products
- has countable products
- has binary products
- has a terminal object
- has pullbacks
- is connected
- is Cauchy complete
- has sequential limits
- has a generator
- is well-copowered
- is well-powered
- has exact filtered colimits
- has filtered colimits
- is locally ℵ₁-presentable
- has coproducts
- is an elementary topos
- is cartesian closed
- is infinitary distributive
- is distributive
- has finite coproducts
- has a strict initial object
- has an initial object
- has a subobject classifier
- has disjoint finite coproducts
- has disjoint coproducts
- is epi-regular
- is finitely cocomplete
- has a cogenerator
- is mono-regular
- is inhabited
- is balanced
- has connected colimits
- has wide pushouts
- has coequalizers
- has countable coproducts
- has binary coproducts
- has pushouts
- has sequential colimits
Non-Properties
Non-Properties from the database
- is not Malcev
- is not skeletal
- does not have a strict terminal object
Deduced Non-Properties*
- is not discrete
- is not additive
- is not preadditive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not trivial
- is not essentially discrete
- is not pointed
- does not have zero morphisms
- is not thin
- is not essentially small
- is not small
- is not essentially finite
- is not finite
- is not left cancellative
- is not self-dual
- is not a groupoid
- is not right cancellative
*This also uses the deduced properties.
Unknown properties
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Special morphisms
- Isomorphisms: bijective maps
- Monomorphisms: injective maps
- Epimorphisms: surjective maps